To develop an equally powerful framework for reasoning about transition structures over posets, it seems natural to study Positive Coalgebraic Logics (henceforth PCL). In fact, work in this direction has already started, see for example [1]}, [2]}. We pursue this work further and present PCL in full generality, i.e. at the same level of generality as its boolean counterpart. Moreover, given the close kinship between the two theories, we will show that the wheel needn't be re-invented every time, and that many BCLs have a canonical positive fragment which inherits useful properties from its boolean parent. The data defining a PCL will be
\({\mathbf {DL}@/^1pc/[rr]^{\mathsf {S^{\prime }}} @(dl,ul)^{L^{\prime }}& \perp &\mathbf {Pos}^{\mathrm {op}}@/^1pc/[ll]^{\mathsf {P}^{\prime }}@(dr,ur)_{(T^{\prime })^{\mathrm {op}}} & & & \delta ^{\prime }: L^{\prime }\mathsf {P}^{\prime }\Rightarrow \mathsf {P}^{\prime }(T^{\prime })^{\mathrm {op}}}\) 
